Comparing to the existing literature, Sakamoto and Yamamoto [1]} were the first to prove the well-posedness of a backward problem like () with smooth data \(u_T\) . More precisely, they considered the case when \(H=L^2(\Omega )\) , with \(\Omega \subset \mathbb {R}^N\)  is a smooth bounded domain, and \(A\)  is a symmetric uniformly elliptic operator with zero Dirichlet boundary condition. Namely, they proved that there exists a constant \(C>0\)  depending on \(T\)  such that for all \(u_T \in H^2(\Omega ) \cap H^1_0(\Omega )\)  the corresponding solution exists and satisfies
\(\Vert u(0,\cdot )\Vert _{L^2(\Omega )} \le C \Vert u(T,\cdot )\Vert _{H^2(\Omega )}.\) 
